Universality out of order

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obscure their causes, […] we may be able to discern a regular movement in it, and that what seems complex and chaotic in the single individual may be seen from the standpoint of the human race as a whole [as] steady and progressive" 9 . Such statistical laws of the human world often take the form of power-law relations, where the exponent is universal, independent of specific details about the systems. Perhaps the most well-known examples are Zipf's law-stating that the frequency of the kth most common word is inversely proportional to k (i.e., having an exponent of -1)-and the Pareto principle of wealth distribution 10 .
In the physical theory of critical phenomena, the meaning of universality came to change in the 1960s and 70 s. The critical exponents, describing the phase transitions, seemed to fall into a few so-called universality classes 11 . Describing how systems change with the length scale-the renormalization group theory -led to a theoretical foundation for this scarcity of critical exponents. Moreover, it motivated statistical physicists to search for universal laws outside of physical systems 12 . It became a prominent example of how universality can appear from emergence-another defining feature of complex systems.

Universality in ranking dynamics
As we have seen so far, ranking dynamics, universality, and criticality are close to the heart of physics-flavored complexity science. It seems obvious that the border area of these ideas would be a fertile ground for discovery. Given that, Iñiguez et al. 4 . extends a surprisingly small collection of studies.
Blumm et al. 13 . took a bona fide physics approach-defining a microscopic model to describe how the fluctuation of the rank of items depends on the rank. They found that the items could belong to one of three phases depending on their fitness and the system's noise level. This situation is notably different from the traditional dynamic critical phenomena in physics, where every unit belongs to the same phase 14 . Rather, it resembles the localization of epidemic phases in heterogeneous networks 15 .
Like Blumm et al. 13 , the recent paper by Iñiguez et al 5 . also builds a microscopic model of list dynamics but differs from the former in two aspects: First, Iñiguez et al 5 . study open lists where (like many real-world rankings) only the top elements are explicitly ranked. Second, they do not assume the ranking is based on a variable associated with the items. The model of Iñiguez et al 5 . consists only of replacement (where an out-of-list item substitutes an element) and displacement (where an element moves within the list and offsets the others). The universality in their model refers to a non-trivial consistency relation that the rates of replacement and displacement must obey, which they furthermore corroborate with empirical data. Details of the displacement dynamics do not matter, and the relation holds whether the jumps in the list follow a fat-tailed distribution or not.

Future outlook
Discovering universal relations of simple observables in list dynamics has more practical benefits than the lofty goals of Kant and others. They give us a more precise language to describe a phenomenon. Thanks to Iñiguez et al., we know the replacement rate is sufficient to explain the state of ranking dynamics. Thanks to Blumm et al., we can describe an item in a ranking just by the phase to which it belongs.
Since rankings are everywhere, so should practical (commercial or otherwise) applications of the theory of ranking dynamics be. While it is hard to speculate about the exact form of such applications, highly predictable, persistent patterns-the statistical laws discussed above-would be valuable as foundations. Ref. 5 also opens several new research directions. First, the role of memory in real-world list dynamics seems to be uncharted territory. One would need more information than about the present to determine the evolution of many types of real-world lists-few people would predict an aged tennis player in decline to bounce back into the top tier. This is in stark contrast to the memory-less models of Refs. 5 and 13 . Second, because of their even more fundamental role for human organization, it would be interesting to study the dynamics of hierarchies (the big caveat, of course, being that there is no universally accepted definition of hierarchy 6 ).
To fulfill the promises of complex systems science, we need to keep discovering fundamental relations governing simple phenomena such as ranking dynamics. Even though not being theories of everything (as was the ambition of some complexity science in the 1990s 12 ), such statistical laws are still the building blocks we need for the physics-type explanations in the social (and life) sciences that Kant and his contemporaries envisioned. Whether or not discoveries like those of Iñiguez et al. can also lead to forms of social organization where such natural laws complement juridical ones-as Enlightenment thinkers also believed-remains to be seen. , not necessarily the one minimizing the number of discordant pairs with respect to the ranking-a so-called "minimum violation ranking"). As typical for real-world rankings of many kinds, we can see that pecking orders of hens are not complete, linear rankings 1 .